How to differentiate y=(2+4x^2)^3?

Expert Answers
justaguide eNotes educator| Certified Educator

We can solve y=(2+4x^2)^3 in two ways. One is use the chain rule

y=(2+4x^2)^3

y' = 3*8x*(2 + 4x^2)^2

=> y' = 24x*( 4 + 16x^4 + 16x^2)

=> y' = 96x + 384x^5 + 384x^3

Else expand (2+4x^2)^3

y=(2+4x^2)^3

=> 8 + 64x^6 + 96x^4 + 48x^2

y' = 384x^5 + 384x^3 + 96x

The derivative of y=(2+4x^2)^3 is 384x^5 + 384x^3 + 96x

giorgiana1976 | Student

Since the given function is composed, we'll apply chain rule to differentiate it.

We'll differentiate with respect to x. First, we'll identify the composed functions, whose final result is y.

dy/dx = (dy/dt)*(dt/dx)

We'll put 2+4x^2 = t

y = t^3

We'll differentiate with respect to t:

dy/dt = d(t^3)/dt

dy/dt = 3t^2

We'll differentiate t with respect to x.

dt/dx = d(2+4x^2)/dx

dt/dx = 8x

dy/dx = 3t^2*8x = 24x*t^2

We'll substitute back t:

dy/dx = 24x(2 + 4x^2)^2

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